US7230692B2ExpiredUtilityPatentIndex 71
Optical apparatus, method of determining position of optical element in optical system, and device manufacturing method
Est. expiryMar 17, 2024(expired)· nominal 20-yr term from priority
G02B 27/0068G03F 7/70258G02B 7/005G03F 7/706
71
PatentIndex Score
9
Cited by
7
References
12
Claims
Abstract
An optical apparatus including an image forming optical system having a movable optical element, and a driving mechanism. A first block obtains a linear evaluation value by normalizing, by a first tolerance, an aberration expressed by a linear function of a position of the movable optical element and normalizing, by a second tolerance, an aberration expressed by a quadratic function of the position out of the aberrations of the optical system. A second block obtains a minimum value of a dummy variable by linear programming. A third block determines a position of the optical element to be moved by the driving mechanism using a value prepared by adding a relaxation amount to the minimum value obtained by the second block and minimizes the weighted sum of the quadratic evaluation values by adjusting the weights assigned to the quadratic evaluation values and a relaxation amount.
Claims
exact text as granted — not AI-modified1. An optical apparatus which includes an image forming optical system having a movable optical element, and a driving mechanism configured to move said optical element, said apparatus comprising:
a first block which obtains a linear evaluation value by normalizing, by a first tolerance, an aberration expressed by a linear function of a position of said movable optical element out of aberrations of said optical system, and a quadratic evaluation value by normalizing, by a second tolerance, a square of a root mean square of wavefront aberration of said optical system expressed by a quadratic function of the position out of the aberrations of said optical system;
a second block which uses a dummy variable t as an upper limit value of the linear evaluation value and obtains a minimum value of the dummy variable t by a linear programming, wherein the linear programming is given by the following formulas,
Minimization f 1 =t
Constraint conditions:
y
0
ih
+
∑
k
=
1
K
c
ihk
x
k
≤
t
-
y
0
ih
-
∑
k
=
1
K
c
ihk
x
k
≤
t
t
≥
0
where
y 0ih : the initial value of the evaluation value of the i-th aberration at the image height h;
x k : the k-th adjustment amount;
c ihk : the degree of influence of the adjustment amount x k of each part on the evaluation value y ih of the image performance; and
a third block which determines a position of said optical element to be moved by said driving mechanism, by a quadratic programming, so as to minimize the quadratic evaluation value under a constraint condition that the upper limit value of the linear evaluation value is a value prepared by adding a relaxation amount d to the minimum value of the dummy variable t that is obtained by said second block,
wherein said third block minimizes the quadratic evaluation value by adjusting the quadratic evaluation values and the relaxation amount based on the prepared value and the quadratic evaluation value having been obtained using the prepared value, wherein the quadratic programming is given by the following formulas,
Minimization:
f
2
=
1
RMS
2
∑
h
=
1
H
W
h
∑
j
=
1
J
α
jh
(
z
0
jh
+
∑
k
=
1
K
b
jhk
x
k
)
2
Constraint conditions:
y
oih
+
∑
k
=
1
n
c
ihk
x
k
≤
t
+
d
-
y
0
ih
-
∑
k
=
1
n
c
ihk
x
k
≤
t
+
d
x
k
≥
0
,
where
w h : weight,
wherein said driving mechanism is configured to move said optical element in accordance with the determined position.
2. An apparatus according to claim 1 , wherein said apparatus is an exposure apparatus which transfers a pattern to a substrate using said optical system.
3. A method of manufacturing a device, said method comprising steps of:
transferring a pattern to a substrate using an exposure apparatus as defined in claim 2 ;
developing the substrate to which the pattern has been transferred; and
processing the developed substrate to manufacture the device.
4. An optical apparatus including an optical system having a movable optical element, and a driving mechanism configured to move said optical element, said apparatus comprising:
a first block which obtains a linear evaluation value by normalizing, by a tolerance, an aberration expressed by a linear function of a position of said movable optical element out of aberrations of said optical system;
a second block which uses a first dummy variable as an upper limit of an absolute value of a Zernike coefficient of a wavefront aberration of said optical system, and obtains a linear evaluation value that approximates root mean square of the wavefront aberration by a linear function of the first dummy variable, wherein the Zernike coefficient is given by the following formula,
z
jh
=
z
0
jh
+
∑
k
=
1
J
b
jhk
x
k
,
where
z 0jh : the initial value of the j-th Zernike coefficient at the image height h,
b jhk : the degree of influence of the adjustment amount x k of each part on the Zernkie coefficient z jh ,
x k : the k-th adjustment amount; and
a third block which uses a second dummy variable t as an upper limit of the linear evaluation values obtained by said first block and said second block, and determines a position of said optical element to be moved by said driving mechanism by a linear programming so as to minimize the second dummy variable, wherein the linear programming is given by the following formulas,
Minimization: f 1 =t
Constraint conditions:
x
k
≥
0
s
jh
≥
0
z
jh
≤
s
jh
-
z
jh
≤
s
jh
y
0
ih
+
∑
k
=
1
K
c
ihk
x
k
≤
t
-
y
0
ih
-
∑
k
=
1
K
c
ihk
x
k
≤
t
1
RMS
∑
j
=
1
J
α
jh
s
jh
≤
t
α
jh
≥
0
s
jh
≥
0
where
c ihk : the degree of influence of the adjustment;
y 0hk : the initial value of the evaluation value of the ith aberration at the image height h,
wherein said driving mechanism is configured to move said optical element in accordance with the determined position.
5. An apparatus according to claim 4 , wherein said apparatus is an exposure apparatus which transfers a pattern to a substrate using said optical system.
6. A method of manufacturing a device, said method comprising steps of:
transferring a pattern to a substrate using an exposure apparatus as defined in claim 5 ;
developing the substrate to which the pattern has been transferred; and
processing the developed substrate to manufacture the device.
7. A method of determining a position of an optical element in an optical apparatus which includes an image forming optical system having the optical element, and a driving mechanism configured to move the optical element, said method comprising:
a first step of obtaining a linear evaluation value by normalizing, by a first tolerance, an aberration expressed by a linear function of a position of the optical element out of aberrations of the optical system, and a quadratic evaluation value by normalizing, by a second tolerance, a square of a root mean square of wavefront aberration of the optical system expressed by a quadratic function of the position out of the aberrations of the optical system;
a second step of using a dummy variable t as an upper limit value of the linear evaluation value, obtaining a minimum value of the dummy variable t by a linear programming, wherein the linear programming is by the following formulas,
Minimization: f 1 =t
Constraint conditions:
y
oih
+
∑
k
=
1
K
c
ihk
x
k
≤
t
-
y
0
ih
+
∑
k
=
1
K
c
ihk
x
k
≤
t
t
≥
0
where
y 0ih : the initial value of the evaluation value of the i-th aberration at the image height h;
x k : the k-th adjustment amount;
c ihk : the degree of influence of the adjustment amount x k of each part on the evaluation value y ih of the image performance; and
a third step of determining a position of the optical element to be moved by the driving mechanism, by a quadratic programming, so as to minimize the quadratic evaluation value under a constraint condition that the upper limit value of the linear evaluation value is a value prepared by adding a relaxation amount d to the minimum value of the dummy variable t that is obtained in the second step,
wherein in the third step, the quadratic evaluation value is minimized by adjusting t the quadratic evaluation values and the relaxation amount based on the prepared value and the quadratic evaluation value having been obtained using the prepared value, and the driving mechanism is configured to move said optical element in accordance with the determined position, and
the quadratic programming is given by the following formulas,
Minimization:
f
2
=
1
RMS
2
∑
h
=
1
H
W
h
∑
j
=
1
J
α
jh
(
z
0
jh
+
∑
k
=
1
K
b
jhk
x
k
)
2
Constraint conditions:
y
0
ih
+
∑
k
=
1
n
c
ihk
x
k
≤
t
+
d
-
y
0
ih
-
∑
k
=
1
n
c
ihk
x
k
≤
t
+
d
x
k
≥
0
,
where
w h : weight, and
wherein the driving mechanism is configured to move the optical element in accordance with the determined position.
8. A method according to claim 7 , wherein the apparatus is an exposure apparatus which transfers a pattern to a substrate using the optical system.
9. A method of manufacturing a device, said method comprising steps of:
moving an optical element in an optical system of an exposure apparatus to a position as determined in accordance with a method as defined in claim 8 ;
transferring a pattern to a substrate using the exposure apparatus of which the optical element is moved in said moving step;
developing the substrate to which the pattern has been transferred; and
processing the developed substrate to manufacture the device.
10. A method of determining a position of an optical element in an optical apparatus which includes an optical system having the optical element, and a driving mechanism configured to move the optical element, said method comprising:
a first step of obtaining a linear evaluation value by normalizing, by a tolerance, and aberration expressed by a linear function of a position of the optical element out of aberrations of the optical system;
a second step of, using a first dummy variable as an upper limit of an absolute value of a Zernike coefficient of a wavefront aberration of the optical system, and obtaining a linear evaluation value which approximates root mean square of the wavefront aberration by a linear function of the first dummy variable, wherein the Zernike coefficient is given by the following formula,
z
jh
=
z
0
jh
+
∑
k
=
1
J
b
jhk
x
k
,
where
z 0jh : the initial value of the j-th Zernike coefficient at the image height h,
b jhk : the degree of influence of the adjustment amount x k of each part on the Zernike coefficient z jh ,
x k : the k-th adjustment amount; and
a third step of, using a second dummy variable t as an upper limit of the linear evaluation values obtained in the first step and the second step, and determining a position of the optical element to be moved by the driving mechanism by a linear programming so as to minimize the second dummy variable, wherein the linear programming is given by the following formulas,
Minimization: f 1 =t
Constraint conditions:
x
k
≥
0
s
jh
≥
0
z
jh
≤
s
jh
-
z
jh
≤
s
jh
y
0
ih
+
∑
k
=
1
K
c
ihk
x
k
≤
t
-
y
0
ih
-
∑
k
=
1
K
c
ihk
x
k
≤
t
1
RMS
∑
j
=
1
J
α
jh
s
jh
≤
t
α
jh
≥
0
s
jh
≥
0
-
y
0
ih
-
∑
k
=
1
K
c
ihk
x
k
≤
t
1
RMS
∑
j
=
1
J
α
jh
s
jh
≤
t
α
jh
≥
0
s
jh
≥
0
where
c ihk : the dgree of influence of the adjustment,
y 0ih : the initial value of the evaluation value of the ith aberration at the image height h,
wherein the driving mechanism is configured to move the optical element in accordance with the determined position.
11. A method according to claim 10 , wherein the apparatus is an exposure apparatus which transfers a pattern to a substrate using the optical system.
12. A method of manufacturing a device, said method comprising steps of:
moving an optical element in an optical system of an exposure apparatus to a position as determined in accordance with a method as defined in claim 11 ;
transferring a pattern to a substrate using the exposure apparatus of which the optical element is moved in said moving step;
developing the substrate to which the pattern has been transferred; and
processing the developed substrate to manufacture the device.Cited by (0)
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